1. Field of the Invention
The present invention relates to a method for determining the behaviour of a viscoelastic material.
It also relates to a method for improving the storage of experimental curves, a method for determining at least one viscoelastic material having predetermined characteristics and apparatus useful for this purpose.
2. Description of the Related Art
Among the properties of viscoelastic materials, those which are mainly associated with their use are the dynamic properties, or the responses to the application of deformations.
At present, in order to be able to select, at the time of use, the material which best satisfies the final dynamic requirements of a desired product, it is necessary to know the dynamic properties of a large number of materials and, therefore, to have first carried out numerous experimental measurements in order to characterize them.
These dynamic properties, however, depend on the temperature and, in the case of some materials, such as for example polymers containing reinforcing fillers or asphalts, also on the deformation applied. Consequently, the characterization of each material requires the acquisition of a large number of experimental values according to variation in the temperature and the deformation applied.
For example, in order to determine the dynamic properties of these materials, generally experimental measurements of the force following deformation cycles in a predetermined temperature range are performed.
Moreover, the storage of the abovementioned data requires a large amount of space or memory depending on whether it is stored on a paper medium or electronic medium. In the case of storage on a paper medium, moreover, a very long time is needed in order to check whether, with the stored experimental data, it is possible to identify a product having the required characteristics, at a certain temperature.
Another drawback inherent in the known methods is that it is possible to identify the product having the required characteristics only if experimental measurements have been performed at the temperature concerned at the moment when said identification is performed.
The inventors of the present invention, therefore, have considered the problem of simplifying the characterization, storage and selection processes which are currently used.
More precisely, considering the experimental curves of a dynamic parameter P as a function of a deformation q applied to a generic test piece of viscoelastic material at a temperature T, the inventors of the present invention have surprisingly discovered that these experimental curves are referable to a summation of exponentials of the type:                               P          ⁡                      (                          q              ,              T                        )                          =                              a            *                          T              2                                +                      b            *            T                    +          c          +                                    ∑                              i                =                1                            ∞                        ⁢                          xe2x80x83                        ⁢                                          (                                                                            d                      i                                        *                                          T                      2                                                        +                                                            e                      i                                        *                    T                                    +                                      f                    i                                                  )                            *                              e                                  -                                      xe2x80x83                                    ⁢                                      q                                          q                      i                                                                                                                              (        I        )            
where
q is the deformation expressed in %;
T is the temperature expressed in Kelvin;
P(q,T) is a dynamic parameter preferably selected from the group comprising the elastic modulus, the viscous modulus, the complex modulus and the loss factor;
a, b, c are characteristic constants which depend on the type of viscoelastic material and the type of dynamic parameter considered;
di, ei, fi are characteristic constants which depend on the effect of the temperature on the dynamic parameter P and on the ith characteristic deformation;
qi is the characteristic deformation at the ith exponential.
The inventors, moreover, have realized that, in order to obtain good approximation of said experimental curves, it is sufficient to perform the abovementioned summation (I) as far as the second term. In such a case, therefore, there are only 11 parameters to be determined for each dynamic parameter P of each viscoelastic material, namely, a, b, c, q1, q2, d1, e1, f1, d2, e2 and f2.
In the case where, on the other hand, it is desired to obtain a greater accuracy, it is sufficient to perform the abovementioned summation (I) as far as the higher term, which ensures the desired accuracy. Obviously, in this case, the number of parameters which must be determined will gradually increase.
Therefore, even though the description which follows is mainly based on carrying out the summation as far as the second term, the person skilled in the art will not have any difficulty in determining the required parameters and in carrying out the abovementioned summation (I) up to any term higher than the second term which ensures the desired accuracy.
For example, in the case of the experimental curves for the elastic modulus Gxe2x80x2, measured at different temperature values T, as a function of a torsion xcex3 applied to a cylindrical test piece consisting of a compound for a tyre containing a reinforcing filler, the inventors have found that these experimental curves are referable, according to the present invention, to the relation:             G      xe2x80x2        ⁢          (              γ        ,        T            )        =            a      *              T        2              +          b      *      T        +    c    +                  (                                            d              1                        *                          T              2                                +                                    e              1                        *            T                    +                      f            1                          )            *              e                  -                      xe2x80x83                    ⁢                      γ                          γ              1                                            +                  (                                            d              2                        *                          T              2                                +                                    e              2                        *            T                    +                      f            2                          )            *              e                  -                      xe2x80x83                    ⁢                      γ                          γ              2                                          
Furthermore, investigating also the experimental curves of the complex modulus G*, viscous modulus Gxe2x80x3
(where |G*|={square root over ((Gxe2x80x2)2+(Gxe2x80x3)2)})
and the loss factor tan xcex4 (where tan xcex4=Gxe2x80x3/Gxe2x80x2) as a function of the torsion xcex3, applied to the test piece, and of the temperature T, the inventors have surprisingly found that they may also be represented by the abovementioned summation of exponentials and that in order to obtain a good approximation, for each of the abovementioned experimental curves, 11 parameters are sufficient, as illustrated above.
Finally, the inventors have even more surprisingly found that the values of the parameters a, b, c, q1, q2, d1, e1, f1, d2, e2 and f2, which have been determined on the basis of at least 5 experimental measurements, carried out at each of at least three temperatures Tx, Ty, Tz, of a dynamic parameter P of a viscoelastic material as a function of a deformation q, may be used to determine the progression of said dynamic parameter P as a function of said deformation q also at a temperature Tw different from Tx, Ty and Tz. Thus, with this method the progression of said dynamic parameter P is determined as a function of said deformation q at a temperature Tw for which no experimental determination was performed.
According to a first aspect thereof, the present invention therefore relates to a method for determining the behaviour of a viscoelastic material at a temperature Tw characterized in that
a) N experimental measurements, where Nxe2x89xa75, of a dynamic parameter P as a function of a deformation q at each of at least three temperatures Tx, Ty, Tz different from Tw are performed;
b) for each of said at least three temperatures Tx, Ty, Tz, the experimental curve which passes through all the points which represent the N values determined experimentally in said step a) is plotted;
c) at least 11 values of said dynamic parameter P and of the associated deformation q distributed along the experimental curves plotted in said step b) at said at least three temperatures Tx, Ty, Tz are chosen;
d) said at least 11 values are inserted in the relation (A):       P    ⁢          (              q        ,        T            )        =            a      *              T        2              +          b      *      T        +    c    +          xe2x80x83        ⁢                  (                                            d              1                        *                          T              2                                +                                    e              1                        *            T                    +                      f            1                          )            *              e                  -                      xe2x80x83                    ⁢                      q                          q              1                                            +          xe2x80x83        ⁢                  (                                            d              2                        *                          T              2                                +                                    e              2                        *            T                    +                      f            2                          )            *              e                  -                      xe2x80x83                    ⁢                      q                          q              2                                          
xe2x80x83so as to determine, for subsequent approximations, values of parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 of said relation (A) which generate the curves which best approximates said experimental curves plotted in step b); and
e) the temperature Tw and the values of said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 obtained in the previous step d) are inserted in said relation (A) in order to determine the progression of said dynamic parameter P as a function of the deformation q at the temperature Tw.
More particularly, during said determination for subsequent approximations of step d)
d1) an arbitrary value for each of said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 is inserted in said relation (A);
d2) by means of said relation (A) at least 11 values of said parameter P are determined as a function of the values of said deformation q indicated in the abovementioned step c);
d3) the differences between the values, thus determined, of said parameter P and the corresponding values of P indicated in the abovementioned step c) are calculated;
d4) the squares of the abovementioned differences are added together;
d5) the value thus obtained of the sum of the squares of the abovementioned differences is stored;
d6) on the basis of the value obtained for said sum, another value is assigned to each of the parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 so as to reduce this sum of the squares of the differences;
d7) the steps from d2) to d6) are repeated until the value of the sum of the squares of the differences is minimized;
d8) the values of said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 which minimized the value of said sum are stored.
These latter values, which are inserted in the abovementioned relation (A), generate the curves which best approximate the experimental curves plotted in the abovementioned step b).
Preferably, at least some of said at least 11 values of the deformation q, used in step d2) to determine P by means of the relation (A), are selected from among those used during the preceding step a). More preferably, all said at least 11 values of the deformation q used in step d2) are chosen from among the values of the deformation q used in said step a).
Preferably, in the abovementioned step c), at least two values of said dynamic parameter P and the associated deformation q for each of said experimental curves are chosen. More preferably, in the abovementioned step c), at least three values of said dynamic parameter P and the associated deformation q for each of said experimental curves are chosen.
Typically, said dynamic parameter P is selected from the group comprising the elastic modulus Pxe2x80x2, the viscous modulus Pxe2x80x3, the complex modulus P* and the loss factor tanxcex4 of said viscoelastic material. Moreover, said deformation q is selected from the group comprising a torsion xcex3, a cutting force xcfx84, a tensioning force and a flexing force.
Advantageously, said at least three temperatures Tx, Ty, Tz are selected in a range of temperatures laying between xe2x88x9255xc2x0 C. and 90xc2x0 C.
When it is required to obtain greater accuracy, the additional parameters required are determined and in place of the abovementioned relation (A) the following summation (I) is performed:                               P          ⁡                      (                          q              ,              T                        )                          =                              a            *                          T              2                                +                      b            *            T                    +          c          +                                    ∑                              i                =                1                            ∞                        ⁢                          xe2x80x83                        ⁢                                          (                                                                            d                      i                                        *                                          T                      2                                                        +                                                            e                      i                                        *                    T                                    +                                      f                    i                                                  )                            *                              e                                  -                                      xe2x80x83                                    ⁢                                      q                                          q                      i                                                                                                                              (        I        )            
where
q is the deformation expressed in %;
T is the temperature expressed in Kelvin;
a, b, c are characteristic constants which depend on the type of viscoelastic material and on the type of dynamic parameter considered;
di, ei, fi are characteristic constants which depend on the effect of the temperature on the dynamic parameter P and on the ith characteristic deformation;
qi is the characteristic deformation at the ith exponential.
According to a second aspect thereof, the present invention relates to a method for improving the storage of at least three experimental curves, each obtained from at least N measured experimental values, where N is equal to at least 5, of a dynamic parameter P of a viscoelastic material as a function of a deformation q at one of at least 3 predetermined temperatures Tx, Ty, Tz, said method being characterized in that it comprises the steps of:
a) choosing at least 11 values of said dynamic parameter P and the associated deformation q distributed along said at least 3 experimental curves obtained at said at least 3 temperatures Tx, Ty, Tz;
b) inserting said at least 11 values in the relation (A):       P    ⁢          (              q        ,        T            )        =            a      *              T        2              +          b      *      T        +    c    +          xe2x80x83        ⁢                  (                                            d              1                        *                          T              2                                +                                    e              1                        *            T                    +                      f            1                          )            *              e                  -                      xe2x80x83                    ⁢                      q                          q              1                                            +          xe2x80x83        ⁢                  (                                            d              2                        *                          T              2                                +                                    e              2                        *            T                    +                      f            2                          )            *              e                  -                      xe2x80x83                    ⁢                      q                          q              2                                          
xe2x80x83so as to determine, for subsequent approximations, values for the parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 of said relation (A) which generate the curves which best approximate said at least 3 experimental curves;
c) storing the values of said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 thus obtained.
As regards the characteristics of said determination for subsequent approximations, said dynamic parameter P, said deformation q and said temperatures Tx, Ty, Tz, reference should be made to that already stated above in connection with the method for determining the behaviour of a viscoelastic material at a temperature Tw.
According to a third aspect thereof, the present invention relates to a method for defining, from among a plurality of viscoelastic materials, at least one viscoelastic material having a dynamic parameter P which, at a preselected temperature Tw and deformation qW, has a predetermined value PW, said method being characterized in that it comprises the steps of
a) determining, for each of said plurality of viscoelastic materials, the value which said dynamic parameter P assumes at said temperature Tw for said deformation qw by means of the relation (A.1)       P    ⁢          (                        q          w                ,                  T          w                    )        =            a      *              T        w        2              +          b      *              T        w              +    c    +          xe2x80x83        ⁢                  (                                            d              1                        *                          T              w              2                                +                                    e              1                        *                          T              w                                +                      f            1                          )            *              e                  -                      xe2x80x83                    ⁢                                    q              w                                      q              1                                            +          xe2x80x83        ⁢                  (                                            d              2                        *                          T              w              2                                +                                    e              2                        *                          T              w                                +                      f            2                          )            *              e                  -                      xe2x80x83                    ⁢                                    q              w                                      q              2                                          
xe2x80x83where, for each of said plurality of viscoelastic materials, the values of the parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 have already been determined beforehand;
b) comparing each of the values of said parameter P thus determined with said value Pw; and
c) defining the viscoelastic materials for which, at the temperature Tw and for the deformation qw, said dynamic parameter P satisfies said predetermined value Pw.
Preferably, said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 of the abovementioned step a) were determined in a manner similar to that already described further above in connection with the first and the second aspect of the present invention.
Moreover, as regards the characteristics of said dynamic parameter P, said deformation qw and said temperature Tw, reference should be made to that stated further above in connection with P, q, Tx, Ty and Tz.
According to a fourth aspect thereof, the present invention relates to an apparatus for processing data relating to a viscoelastic material having a predetermined dynamic parameter P, said apparatus comprising
a) an input for entering N experimentally determined values, where Nxe2x89xa75, of said dynamic parameter P as a function of a deformation q at each of at least three temperatures Tx, Ty, Tz;
b) a processing unit connected to said input, said processing unit being capable of
b1) plotting, for each of said at least three temperatures Tx, Ty, Tz, the experimental curve which passes through all the points which represent said N experimental values entered via said input;
b2) choosing at least 11 values of said dynamic parameter P and the associated deformation q distributed along the experimental curves plotted in step b1) at said at least three temperatures Tx, Ty, Tz, and
b3) inserting said at least 11 values in the relation (A)       P    ⁢          (              q        ,        T            )        =            a      *              T        2              +          b      *      T        +    c    +          xe2x80x83        ⁢                  (                                            d              1                        *                          T              2                                +                                    e              1                        *            T                    +                      f            1                          )            *              e                  -                      xe2x80x83                    ⁢                      q                          q              1                                            +          xe2x80x83        ⁢                  (                                            d              2                        *                          T              2                                +                                    e              2                        *            T                    +                      f            2                          )            *              e                  -                      xe2x80x83                    ⁢                      q                          q              2                                          
xe2x80x83so as to determine, for subsequent approximations, values for the parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 of said relation (A) which generate the curves which best approximate said experimental curves plotted in step b1);
c) a memory for storing the values of the parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 thus determined.
Advantageously,
via said input a predetermined temperature Tw different from said at least three temperatures Tx, Ty, Tz is also entered; and
said processing unit is also capable of determining, by means of said relation (A), in which the values of said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 stored in said memory are inserted, the progression of said dynamic parameter P as a function of said deformation q at said temperature Tw.
Preferably, said apparatus also comprises an output for providing said progression of said dynamic parameter P as a function of the deformation q at the temperature Tw.
As regards the characteristics of said determination for subsequent approximations, said dynamic parameter P, said deformation and said temperatures Tx Ty, Tz and Tw, reference should be made to that already stated further above.
According to a fifth aspect thereof, the present invention relates to an apparatus for processing data relating to a plurality of viscoelastic materials having a dynamic parameter P, said apparatus comprising
an input for entering a value Pw which represents a reference value for said dynamic parameter P at a preselected temperature Tw for a preselected deformation qw;
a processing unit capable of
i. determining, for each of said plurality of viscoelastic materials, the value which the associated dynamic parameter P assumes at said temperature Tw and at said deformation qw by means of the relation (A.1)       P    ⁢          (                        q          w                ,                  T          w                    )        =            a      *              T        w        2              +          b      *              T        w              +    c    +          xe2x80x83        ⁢                  (                                            d              1                        *                          T              w              2                                +                                    e              1                        *                          T              w                                +                      f            1                          )            *              e                  -                      xe2x80x83                    ⁢                                    q              w                                      q              1                                            +          xe2x80x83        ⁢                  (                                            d              2                        *                          T              w              2                                +                                    e              2                        *                          T              w                                +                      f            2                          )            *              e                  -                      xe2x80x83                    ⁢                                    q              w                                      q              2                                          
xe2x80x83where, for each of said plurality of viscoelastic materials, the values of the parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 have already been determined beforehand; and
ii. comparing the value of said parameter P thus determined with said value Pw; and
an output for providing, from amongst said plurality of viscoelastic materials, the viscoelastic materials for which, at the temperature Tw and at the deformation qw, said dynamic parameter P satisfies said value Pw.
According to a first embodiment, said apparatus also comprises a memory in which said values of said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 are stored.
According to a second embodiment, said values of said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 are entered into the apparatus via said input.
As regards the determination of said values of said parameters a, b, c, q1, q2, d1, e1, f1, d2, e2, f2 and the characteristics of the dynamic parameter P, deformation qw and temperature Tw, reference should be made to that already stated further above.